Theory of the Interaction of Vacancies with Stress Fields in Metals. I. Derivation of Basic Equations

Abstract

A theory for the effect of stresses on vacancy concentration in simple metals is presented in which the variation of mechanical energy, electronic energy, and vibration frequencies with stress is taken into account. The vacancy is treated as a point of dilatation and a local soft spot in the crystal with the result that the mechanical energy consists of a term representing the work done by the volume of relaxation against the internal pressure, plus an additional term which is quadratic in the stresses. The electronic energy term is computed from the quasifree electron model based on phase‐shift analysis, giving a term proportional to the dilatation. The variation of atomic frequencies with strain is treated by a generalized Gruneisen model which shows that the entropy of vacancy formation is proportional to the dilatation. The internal work and electronic terms in the energy of interaction of a vacancy with a stress field decreases the vacancy concentration in dilated regions and increases it in compressed regions relative to the concentration in unstrained metal. The quadratic part of the mechanical energy, which comes from the softening of the crystal by the vacancy, always acts to increase the vacancy concentration in regions of high stress, regardless of the sign of the stress field. The entropic contribution, arising from the vibration frequencies, opposes the internal work and electronic terms, decreasing the vacancy concentration in compressed regions and increasing it in dilated regions. The final result is that the interaction energy of a vacancy with a stress field is a function whose magnitude decreases linearly with temperature.